Emulating human brain - with analogous NN chips
I'm not sure about "emulating the brain" per-se, but in a more general sense there has been some thought given to using analog computing for AI/ML.  It seems clear that analog computers do have certain advantages over digital computers.  For one, they can (depending on the application) be faster, albeit at the cost of some loss of precision.  But that's OK, because I don't think anybody believes the human brain is calculating floating point math using digital computing techniques either.  The human brain appears, at least superficially, to be largely probabilistic and able to tolerate some "slop" numerically.  

The downside to analog computers, as I understand it, is that they're not as flexible... you basically hardwire a circuit to do one specific "thing" and that's really all it can do.  To change the "programming" you have to literally solder in a new component! Or, I suppose, adjust a potentiometer or adjustable capacitor, etc.  Anyway, the point is that digital computers are supremely flexible, which is one big reason they came to dominate the world.  But I can see where there could be room for going analog for discrete functions that make up some or all of an intelligent system.

As for research in the area, you might look into whatever DARPA was / is doing. There was an article in Wired a while back, talking about some DARPA initiatives related to analog computing.